User blog:B1mb0w/The Big number function
'The Big number function' The Big number function is a very fast growing function. It's growth rate is well beyond \(f_{LVO}(n)\). The Big number function is a pair of functions \(B()\) and \(g()\) which use this simple rule set: \(B(n) = B(0,n) = n + 1\) \(B(a + 1, n) = B^n(a,n_*)\) \(B(g(0), n) = B(n,n)\) and other instances of \(n\) can be substituted with \(g(0)\) \(g(c + 1) = g(0, c + 1) = B^{g©}(g©_*,g©)\) and \(g(1, 0_{+ 1}) = g^{g(1, 0_{d})}(1_*, 0_{d})\) \(g(b + 1, 0) = g^{g(b,0)}(b,0_*)\) \(g(b, c + 1) = B^{g(b,c)}(g(b,c)_*,g(b,c))\) and \(g() = g_0()\) \(g_{a + 1}(0) = g_a(1, 0_{g_a(0)})\) \(g_a(c + 1) = B^{g_a©}(g_a©_*,g_a©)\) \(g_a(b + 1, 0) = g_a^{g_a(b,0)}(b,0_*)\) \(g_a(b, c + 1) = B^{g_a(b,c)}(g_a(b,c)_*,g_a(b,c))\) \(g_a(1, 0, 0) = g_a^{g_a(1, 0)}(1_*, 0)\) and \(B(1, 0, n) = B(g_{g(0)}(0),n)\) \(B(a + 1, 0, n) = B(a,g_{g(0)}(0),n)\) 'Notation Explained' I use notation that is not in general use, but I find helpful. They are the \(*\) and parameter subscript brackets. The \(*\) notation is used to explain nested functions. For example: \(M(a) = M(a)\) \(M^2(a) = M(M(a))\) then let \(M^2(a,b_*) = M(a,M(a,b))\) \(M^2(a_*,b) = M(M(a,b),b)\) Parameter subscript brackets are useful for functions with many parameters. For example: \(M(a,0_{1}) = M(a,0)\) \(M(a,0_{3}) = M(a,0,0,0)\) \(M(a,b_{2}) = M(a,b_1,b_2)\) \(M(a,0_{2},b_{3},1) = M(a,0,0,b_1,b_2,b_3,1)\) 'Growth Rate of the Big number function ... to \(\Gamma_0\)' The Big number function behaves like the FGH function up to a point. Also refer to more detailed explanations in my previous blog The T-Rex Function: \(B^h(g,n_*) = f_g^h(n)\) \(B(g(0),n) = f_{\omega}(n)\) \(B(B(1,g(0)),n) = f_{\omega.2}(n)\) \(B(B(3,g(0)),n) = f_{\varphi(1,0)}(n)\) \(B(B(g(0),g(0)),n) \approx f_{\varphi(\omega,0)}(n)\) \(B(g(1),n) = B(B^{g(0)}(g(0)_*,g(0)),n) > B(B^{g(0)}(3_*,g(0)),n) \approx f_{\varphi(1,0,0)}(n) = f_{\Gamma_0}(n)\) 'Growth Rate ... to small Veblen ordinal (svo)' The Big number function will easily reach and surpass the small Veblen ordinal (svo): \(B(g(1),g(1)) > \varphi(1,0,1)\) \(B(B(g(1)),g(1)) = B^{g(1)}(g(1),g(1)_*) > \varphi^2(1,0,0_*)\) \(B(B^{g(0)}(g(1)),g(1)) > \varphi(1,1,0)\) \(B^3(g(1)_*,g(1)) \approx \varphi(1,2,0)\) \(g(2) = B^{g(1)}(g(1)_*,g(1)) \approx \varphi^2(1,0_*,0)\) \(g(3) \approx \varphi^3(1,0_*,0)\) \(g^2(0) \approx \varphi(2,0,0)\) \(g(B^2(g(0))) \approx \varphi^2(2,0_*,0)\) \(g(B(1,g(0))) \approx \varphi(3,0,0)\) \(g^2(1) = g(g(1)) \approx \varphi^2(1_*,0,0)\) \(g^{g(0)}(1) \approx \varphi(1,0,0,0)\) \(g^{B(g(0))}(1) \approx \varphi^2(1,0_*,0,0)\) \(g^{B(1,g(0))}(1) \approx \varphi(2,0,0,0)\) \(g(1,0) = g^{g(1)}(1) > \varphi(\varphi(1,0,0),0,0,0)\) \(g(1,1) > \varphi^2(\varphi(1,0,0),0,0_*,0)\) \(g(1,g(0)) > \varphi(\varphi(1,0,0),1,0,0)\) \(g^2(1,0_*) = g(1,g(1,0)) > \varphi^2(\varphi(1,0,0),0_*,0,0)\) \(g^{g(0)}(1,0_*) > \varphi(\varphi(1,0,0) + 1,0,0,0)\) \(g(2,0) = g^{g(1,0)}(1,0_*) > \varphi^2(1_*,0,0,0)\) \(g(g(0),0) > \varphi(1,0,0,0,0) = \varphi(1,0_{4})\) \(g(1,0,0) > \varphi(1,0_{5})\) \(g(1,0_{n/2}) > \varphi(1,0_{n}) = svo = \vartheta(\Omega^\omega)\) \(g_1(0) = g_0(1,0_{g_0(0)}) = g(1,0_{n}) > \varphi(1,0_{n.2}) = \vartheta(\Omega^{\omega.2})\) 'Growth Rate ... to large Veblen ordinal (LVO) and beyond' The Big number function is one of the Fastest Computable functions where: \(g(0) \approx \omega = \vartheta(0)\) \(B(3,g(0)) \approx \epsilon_0 = \varphi(1,0) = \vartheta(1)\) \(g(1) \approx \Gamma_0 = \varphi(1,0,0) = \vartheta(\Omega^2)\) \(g(1,0_{n/2}) > svo = \vartheta(\Omega^\omega)\) TREE(n) function \(≥ f_{\vartheta(\Omega^\omega\omega)}(n)\) \(g_1(0) > \vartheta(\Omega^{\omega.2})\) \(g_2(0) = g_1(1,0_{g_1(0)}) > \vartheta(\Omega^{\omega.6})\) \(g_{n/2}(0) > \vartheta(\Omega^{\omega^2})\) \(B(1,0,n) = B(g_{g(0)}(0),n) > f_{\vartheta(\Omega^{\omega^4})}(n)\) \(B(g_0(0),0,n) > f_{\vartheta(\Omega^{\omega^{\omega^2}})}(n)\) \(B(g_1(0),0,n) > f_{\vartheta(\Omega^{\Omega})}(n)\) Large Veblen ordinal \(LVO ≥ f_{\vartheta(\Omega^\Omega)}(n)\) \(B(1,0,0,n) = B(g_{g(0)}(0),0,n) > f_{\vartheta(\Omega^{\Omega^2})}(n)\) \(B(1,0_{g(0)},n) = B(1,0_{n},n) \approx f_{\vartheta(\varepsilon_{\Omega+1})}(n)\) Bird's H(n) function \(\approx f_{\vartheta(\varepsilon_{\Omega+1})}(n) = f_{\vartheta(\Omega\uparrow\uparrow\omega)}(n)\) Bird's U(n) function \(\approx f_{\vartheta(\Omega_\omega)}(n)\) Bird's S(n) function (new definition) \(\approx f_{\vartheta(\Omega_\Omega)}(n)\) 'Some Identities' Some Big number function identities are: \(B(B(B(a,b)),b) > B(B(a,b),B(a,b))\) because \(B(B(B(a,b)),b) = B^b(B(a,b),b_*) = B(B(a,b),B^{b-1}(B(a,b),b_*))\) and \(B^{b-1}(B(a,b),b_*) > B(B(a,b),b) > B(a,b)\) 'Further References' Further references to relevant blogs can be found here: User:B1mb0w Category:Blog posts